Hurtling Towards A Cliff

As budget negotiations get underway with the threat of sequestration looming, it’s worth recalling a basic lesson from game theory.

Consider two parties in the same vehicle speeding towards a cliff. The one who concedes, i.e. chickens out and steers the car out of danger, is the loser. Winning is better than losing but either is better than driving off the cliff. Finally, time is valuable: if you are going to concede, you prefer to do it earlier rather than later. Still you are prepared to wait if you expect your rival will concede first.

In equilibrium of this game, unless someone concedes right away there is necessarily a positive probability that they will go over the cliff.

The proof is simple. Consider player 1 and suppose his strategy is not to concede immediately. Then we will show 1’s strategy is such that if 2 never concedes there is a positive probability that 1 will also never concede and they will drive off the cliff together. To prove it, suppose the contrary: that 1’s strategy will eventually concede with probability 1 (if 2 doesn’t concede first). If that is 1’s strategy then 2’s best reply is to wait for 1 to concede. In equilibrium 2 will play such a strategy and the outcome will therefore be that 1 is the loser with probability 1. But if 1 is going to be the loser for sure anyway he should have conceded immediately. That’s a contradiction. We have shown that if 1 does not concede immediately then his strategy will allow the car to drive off the cliff with positive probability. The exact same argument applies to 2. Thus in equilibrium, if the game begins without an immediate concession there is a positive probability they will plunge from the cliff.

@Danny: if you stick with the game longer, then your commitment to your path is obviously greater (for any outside observer). Thus, if you subsequently concede, you’ve shown that you really can’t be trusted; even when all signals are, “I’m fully committed!”, you don’t really mean it. Thus, the best face-saving options are:
1) Concede quickly (for some other gain)
2) Never concede.

There’s “face to be saved” either way; it just diminishes the longer you play the game.

Sojourner – I\’m in a hopeful mood today as it apepars Obama goes over the top. I\’m sharing my positive vibes in the hopes they reach Team Clinton…Regina – Bill & Hillary have true fans within the Democratic Party. Let them loose in those communities to campaign for Obama. Bill also has a great roladex that needs to be opened up in favor of Obama…

This confuses a war of attrition with a game of hawk/dove (chicken). In the latter, the Nash Equilibrium always has one player conceding.

Your underlying assumption here is that the ex-ante expected losses arising going over the cliff are equivalent (or less than) to those resulting from conceding; this, presumably is not case. If the “loss” from conceding is smaller than the loss of going over the cliff then not concede/not concede is sub-optimal – and one player will eventually concede.

I am not assuming that. You are correct that in a *pure-strategy equilibrium* one party must concede. But they statement covers that. In a pure equilibrium one party concedes *immediately*

If there is no immediate concession then in equilibrium each party must be randomly conceding at each point in time. And at each point in time there is a positive probability that neither concedes. Including at the end.

Remember, it’s not really a two player game since Congress has 535 members — how about using evolutionary game theory (Maynard Smith and Price, “the logic of animal conflict”) to model this game — that is, imagine a population of players, including irrational (ie ideological) types and finding the ESS or evolutionarily stable strategy

Surely neither would concede until they had reached what they regarded as the “last moment”, at which they should concede.

Different assessments of that last moment could lead to a mistake, but it is always going to be worth waiting until that last moment (or last moment minus an estimated margin of error, perhaps) before conceding, in the hope that the other person acts illogically.

I don’t question the game theory, but It’s not the case that both parties lose equally by going off of the fiscal cliff. Republicans are more likely to get blamed and also lose out in policy terms. The consensus in the political blogosphere seems to be that while avoiding the cliff is preferable, Obama and the Democrats can actually strengthen their bargaining position by waiting until January. I would be interested in seeing an analysis of how the game is played given that players 1 and 2 do not face equal consequences.

Note that this assumes that both sides have the same perceptions of the costs/benefits (i.e., here, that both sides accept that the GOP will shoulder greater blame). Given the amount of spin on the election seen on Fox and the conservative news outlets, I’d say that the conservatives believe that Obama will shoulder the blame (and that, if he doesn’t, they’ve taken the “principled stand” and, in good martyr fashion, borne their cross admirably – Christ-like imagery thoroughly intentional, as playing the “martyr oppressed for principled faith and beliefs” is a strong motif in conservative, esp. socially conservative, circles, from what I’ve seen).

I think the modeling would be improved if we add some dynamics and asymmetric information: a) the expected value of falling off the cliff increases as t—>T, therefore, chicken out payoff becomes better off symmetrically. b) players may know that each other will chicken out, but that does not matter, what matters is the t(1) critical time of chicken out -hence no solution at time t(0)-, which is determined by its risk aversion (unknown to each other). Now, having said that, the big limitation to apply this game to the Congress case is that in this case there is life after the cliff -no elections in 2013-, thus, it is not only rational to wait until Dec 31st for final solution, but even right after, during the first few months of the implementation of the cliff. Bottom line: buy the index on Dec 31st to hold it for a couple of months.

[…] the endpoint of the renegotiation is the benchmark with which to compare proposed current deals. As in Jeff’s earlier post, the negotiations would take the form of a “war of attrition” and we could end up in […]